If the hard reset technique doesn't work, you may be out of luck. I talked extensively with Apple Support about it as well, and they have no further documentation or troubleshooting techniques. Mine would be charged (tested - the battery had a full voltage) and would turn off after only a few seconds of listening at any volume level. The amount of time would vary based on how long it had been after a hard reset.
The "repair cost" of Apple makes sense because they probably don't even attempt repair. The circuit board inside the headphones all but requires damage to be disassembled, and Apple would not return a damaged product.
If you're curious, take a look at the teardown guide that I will be making in the next few days.